Abstract
The iterated Cauchy problem under consideration is
Here {A 1,..., An} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachA k is a (C 0) semigroup generator, this semigroup is of class (C 0) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachA k generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC −1f→0; hereC is bounded and injective. When theA k are commuting generator withA k-Aj injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form\(u = \sum\nolimits_{i = 1}^n {u_i } \) whereu i is a solution ofu′ i=Aiui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.
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Research of first author supported by an NSF grant.
Research of second author supported by an Ohio University Research Grant.
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Goldstein, J.A., de Laubenfels, R. & Sandefur, J.T. Regularized semigroups, iterated Cauchy problems and equipartition of energy. Monatshefte für Mathematik 115, 47–66 (1993). https://doi.org/10.1007/BF01311210
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DOI: https://doi.org/10.1007/BF01311210